Does anyone know how it has become the standard to express Lie algebras in fraktur? I'd also like to know how it's established for each era and region, not only the origin.

It doesn't seem that Pontrjagin's book, "Topological Groups", published in USSR in 1946 and the second revision in 1954, used fraktur. (It's a mere guess, judging from the Japanese translation which didn't employ fraktur. FYI, it uses Fraktur for Classical Lie groups(!), and their Lie algebras in Roman.)

The wikipedia page on Lie algebra didn't mention at all that they're in fraktur, so I wrote it just now. Thanks beforehand.

EDIT: The original quetion didn't say "I'd also like to know how it's established for each era and region, not only the origin.", and I added it after the first answer.

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To reiterate what is said in the MO post (thanks Cocopuffs), Lie theory was birthed (mostly) in Germany. Sophus Lie worked closely with Felix Klein for a quite a long time. Lie's student Friedrich Engel was German. Wilhelm Killing was German. So the early fathers of Lie theory were pretty much all German (or working in Germany).